Question

Use the quadratic formula to solve. Express your answer in simplest form.
6, x, squared, minus, 5, x, minus, 2, equals, minus, 6, x, squared
6x
2
−5x−2=
−6x
2

Answer 1

To solve the equation \( 6x^2 - 5x - 2 = -6x^2 \), we first need to bring all terms to one side of the equation.

1. Add \( 6x^2 \) to both sides:

\[
6x^2 + 6x^2 - 5x - 2 = 0
\]

This simplifies to:

\[
12x^2 - 5x - 2 = 0
\]

2. Now we can apply the quadratic formula, which is given by:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Here, \( a = 12 \), \( b = -5 \), and \( c = -2 \).

3. We first calculate the discriminant (\( b^2 - 4ac \)):

\[
b^2 - 4ac = (-5)^2 - 4(12)(-2) = 25 + 96 = 121
\]

4. Now we substitute \( a \), \( b \), and the discriminant into the quadratic formula:

\[
x = \frac{-(-5) \pm \sqrt{121}}{2(12)}
\]

This simplifies to:

\[
x = \frac{5 \pm 11}{24}
\]

5. Now we calculate the two possible values for \( x \):

For the positive case:

\[
x = \frac{5 + 11}{24} = \frac{16}{24} = \frac{2}{3}
\]

For the negative case:

\[
x = \frac{5 - 11}{24} = \frac{-6}{24} = -\frac{1}{4}
\]

6. Therefore, the solutions to the equation \( 12x^2 - 5x - 2 = 0 \) are:

\[
x = \frac{2}{3} \quad \text{and} \quad x = -\frac{1}{4}
\]

These are the solutions in simplest form.